Economics Letters 101 (2008) 249–252
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Economics Letters
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o n b a s e
Bargaining with many players: A limit result ☆
Klaus Kultti a, Hannu Vartiainen b,c,⁎
a
b
c
University of Helsinki, Finland
Turku School of Economics, Rehtorinpellonkatu 3, 20500 Turku, Finland
Yrjö Jahnsson Foundation, Finland
a r t i c l e
i n f o
Article history:
Received 30 January 2007
Received in revised form 13 August 2008
Accepted 19 August 2008
Available online 26 August 2008
a b s t r a c t
We provide a simple characterization of the stationary subgame perfect equilibrium of an alternating offers
bargaining game when the number of players increases without a limit. Core convergence literature is
emulated by increasing the number of players by replication. The limit allocation is interpreted in terms of
Walrasian market for being the first proposer.
© 2008 Elsevier B.V. All rights reserved.
Keywords:
Non-cooperative bargaining
Stationary equilibrium
Replication
Walrasian market
JEL classification:
C72
C78
1. Introduction
We study an n-player alternating offers bargaining game where
the players try to agree on a division of a pie. Time proceeds in discrete
periods to infinity, player 1 starts the game, and the proposer in any
period is the player who first rejected the offer of the previous period.
We are interested in what happens when the number of players
increases. Our way of increasing the population parallels the core
convergence literature as we replicate the situation so that while
the number of players is increased the size of the pie increases
proportionally: each replica of players brings in a new pie to the pool
of shareable pies. This could reflect matters e.g. when similar nations
group together as a federation.
Having a large set of players is attractive since in the limit almost all
players act as responders; only one player enjoys the first proposer
advantage and hence, as the number of replicas becomes large, the
solution becomes almost distortion-free. We show that in the limit the
unique stationary subgame perfect equilibrium has a simple characterization in terms of a single replica's preferences. Finally, the
resulting single replica outcome has an attractive Walrasian interpretation: the unique equilibrium in a market where the first proposer
right is sold to a single replica of bargainers induces the same outcome.
☆ Comments of a referee greatly improved the paper. We also thank Hannu Salonen
for discussions.
⁎ Corresponding author. Tel.: +358 40 7206808; fax: +358 9 605002.
E-mail address: hannu.vartiainen@tse.fi (H. Vartiainen).
0165-1765/$ – see front matter © 2008 Elsevier B.V. All rights reserved.
doi:10.1016/j.econlet.2008.08.022
The primitive of our model are the time preferences á la Fishburn
and Rubinstein (1982). This approach does not make assumptions
concerning the concavity of utility functions.1 Under similar assumptions, Kultti and Vartiainen (2007) show that the stationary
equilibrium outcome converges to the Nash-bargaining solution
when the length of the time period goes to zero. Since no additional
assumptions are made on the utility representations, this is an
extension of Binmore, Rubinstein and Wolinsky (1986). We now show
that the limit outcome under replication (but fixed time interval)
converges to a well defined solution also, but different from the Nash
solution.2
2. The model
A pie of size X N 0 is to be divided among the set N = {1, 2, …n} of
players. The set of divisions of the pie is
Sn ð X Þ ¼ x a ℝn : ∑ni¼1 xi X; xi 0; for all i :
Let us write x = (x1,…,xn) and x-i = (x1,…, xi-1, xi + 1,…, xn).
1
Our assumptions about preferences are weaker than, for instance, in Kirshna and
Serrano (1996). As their (unique) equilibrium is stationary, our results can be
interpreted as an extension of theirs.
2
One could also study what happens when the size of the cake is kept fixed and the
number of players is increased. Then there is convergence to the Nash-bargaining
solution because the utility frontier becomes practically linear. The same reasoning
applies when the size of the cake is increased while keeping the number of players
fixed.
250
K. Kultti, H. Vartiainen / Economics Letters 101 (2008) 249–252
A1–A5 hold throughout the paper. By A2, the Pareto-optimal
divisions at any date are given by
that in all subgames each player must get zero. Otherwise there would
be a subgame where some offer y = (y1, …, yn) is accepted. Because of
stationarity this offer is accepted in every subgame. In particular,
player 1 can deviate in the first period and offer y = (y1,…,yn). This is a
profitable deviation and constitutes a contradiction with the assumption that there is a stationary SPE where the game never ends.
Assume next that there is a stationary SPE where an offer x(i) by
some player i a {1, 2,…,n}, is not accepted immediately. Denote by z(i)
the equilibrium outcome in a subgame that starts with an offer x(i) of
player i. But now player i could offer z(i) instead of x(i); everyone else
would accept the offer as in the stationary equilibrium acceptance
depends only on the offer.
Thus, in any equilibrium, i(t)'s offer x(i(t)) = (xj(i(t)))jaN is accepted
at stage t a {0, 1, 2, …}. In stationary equilibrium the time index t can
be relaxed from x(i(t)). An offer x by i is accepted by all j ≠ i if
P n ð X Þ ¼ x a Sn ð X Þ : ∑ni¼1 xi ¼ X :
xj ðiÞ vj xj ð jÞ ; for all j≠i:
For each i there is a function vi: [0, X] →[0, X], defining the present
consumption value of xi in date 1:
Player i's equilibrium offer x(i) maximizes his payoff with respect
to constraint (5) and the resource constraint. By A3, all constraints in
(5) and the resource constraint must bind. That is,
The players' preferences over divisions and timing constitute
the primitive of the model. The pie can be divided at any point of
time T = {0, 1, 2,…}. Let division 0 = (0,…,0) serve as the reference
point, and let (complete, transitive) preferences over S × T satisfy, for
all x, y a S, for all i a N and for all s, t a T, the following properties
(Fishburn and Rubinstein, 1982; Osborne and Rubinstein, 1990, Ch. 4):
(x, t) ⪰i (0, 0)
(x, t) ⪰i (y, t) if and only if xi ≥ yi.
If s N t, then (x, t) ⪰i (x, s), with strict preference if xi N 0.
If (x k, tk) ⪰i (yk, sk) for all k = 1, …, with limits (xk, tk) →(x, t) and
(yk, sk) →(y, s) then (x, t) ⪰i (y, s).
A5. (x, t) ⪰i (y, t + 1) if and only if (x, 0) ⪰i (y, 1), for any t a T.
A1.
A2.
A3.
A4.
ð y; 0Þfi ðx; 1Þ if vi ðxi Þ ¼ yi ; for all x; y a Sn ð X Þ:
ð1Þ
Fishburn and Rubinstein (1982) show that given A1–A5, vi(·) is
continuous and increasing on [0, X].
We assume that the loss of delay increases in the share of the pie.
¼
1
N1; for all xi 0:
viVðxi Þ
ð2Þ
This property will be used when we prove the existence of a
stationary equilibrium.
3. The game
Given N and X, we focus on a unanimity bargaining game Γ N(X)
defined as follows: At any stage t a {0, 1, 2, …},
• Player i(t) a N makes an offer x a Sn (X). Players j ≠ i (t) accept or
reject the offer in the ascending order of their index.3
• If all j ≠ i (t) accept, then x is implemented. If j is the first who rejects,
then j becomes i(t + 1).
• i(0) = 1.
We focus on the stationary subgame perfect equilibria, simply
equilibria or SPE in the sequel, of the game, where:
1. Each i a N makes the same proposal x(i) whenever he proposes.
2. Each i's acceptance decision in period t depends only on xi that is
offered to him in that period.
We now characterize equilibria (see Krishna and Serrano, 1996).
Proposition 1. x is a stationary equilibrium outcome of Γ
only if x = (x1 + d, x2, …, xn), for the x such that
x ¼ vi P
x i þ d ; for all iaN;
Pi
n
∑ xi ¼ X−d
i¼1
N
(X) if and
ð7Þ
Since player i's acceptance decision is not dependent on the name
of the proposer, there is xi N 0 such that xi( j) = xi for all j ≠ i. By Eq. (6),
xj(i) b xj(j) for all j. Hence there is d N 0 such that
∑ni¼1 x i ¼ X−d:
ð8Þ
By Eqs. (6) and (8), x and d do meet Eqs. (3) and (4). Since 1 is the
first proposer, the resulting outcome is x(1) = (x1 + d, x2,…, xn).
If: Let x and d meet Eqs. (3) and (4). Construct the following
stationary strategy: Player i always offers x-i and does not accept less
than xi. Player i's offer y is accepted by all j ≠ i only if
x k ¼ vj x j þ d ; for all j≠i:
ð9Þ
yj vj X−∑k≠j Since vj is increasing, and since
x ¼ vj P
x j þ d ; for all j≠i;
Pj
i's payoff maximizing offer to each j is x j.
5
Thus, to find a stationary equilibrium it is sufficient to find x and d
that meet Eqs. (3) and (4).
By Eq. (2), v−i 1(xi) − xi is a continuous and monotonically increasing
function. Thus, the function ei(·) such that
ei ðxi Þ :¼ v−1
i ðxi Þ−xi ; for any xi 0;
ð10Þ
is continuous and monotonically increasing.
Define ēi a (0, ∞] by
ð3Þ
sup ei ðxi Þ :¼ e i :
ð4Þ
Since ei(·) is continuous and monotonically increasing, also its
inverse
Proof. Only if: In a stationary SPE the game ends in finite time.
Assume that it never ends. Then each player receives zero. This means
3
n
i¼1
That is,
dxi
ð6Þ
and
∑ xi ð jÞ ¼ X; for all j:
A6. xi − vi(xi) is strictly increasing and differentiable.
dv−1
i ðxi Þ
xj ðiÞ ¼ vj xj ð jÞ ; for all j≠i;
ð5Þ
The order in which players' response to a proposal does not affect the results.
xi 0
xi ðeÞ :¼ e−1
i ðeÞ; for all ea½0; e i ;
is continuous and monotonically increasing in its domain [0, ēi].
Condition (10) can now be stated in the form
xi ðeÞ ¼ vi ðxi ðeÞ þ eÞ; for all ea½0; e i :
ð11Þ
K. Kultti, H. Vartiainen / Economics Letters 101 (2008) 249–252
Proposition 2. There is a unique stationary equilibrium of Γ N(X).
Proof. By A1 and A3, xi(0) = 0. Since, for all i, x−i 1 (·) is a monotonically
increasing function on ℝþ having its supremum at ē, it follows that
lime→ēi xi(e) = ∞. Thus, since ∑in= 1 xi(e) + e is a continuous function of e
on [0, ēi] ranging from 0 to ∞, there is, by the Intermediate Value
Theorem, a unique d N 0 such that
n
∑ xi ðdÞ ¼ X−d:
251
∞
Lemma 4. Let {x1·(kt)}∞
t = 1 be a convergent subsequence of {x1·(kt)}k = 1 such
that x1·(kt)→y. Then ∑ni = 1yi = 1.
Proof. By (2), and the continuity of vi,
yi ¼ limt P
x1i ðkt Þ
x1i ðkt Þ þ dðkt ÞÞ
¼ limt vi ðP
x1i ðkt Þ þ limt dðkt ÞÞ
¼ vi ðlimt P
¼ vi ðyi þ limt dðkt ÞÞ:
i¼1
By Eq. (11)0, the pair (x(d), d) meets Eqs. (3) and (4).
5
For later purposes, we now identify a property of the players
preferences. Let X = 1.
Lemma 1. There is unique y⁎ and d⁎ N 0 such that
yTi ¼ vi yTi þ dT ; for all i ¼ 1; ::; n;
n
∑ yi ¼ 1:
i¼1
ð12Þ
ð13Þ
Proof. Let ei(·) be defined as in Eq. (10). Since ei(·) = x-i 1(·) is a
monotonically increasing function from ℝþ to[0, ēi] with ei(0) = 0,
there is a unique d⁎ a (0, ēi] such that ∑in= 1 xi(d⁎) = 1. By (11), xi(d⁎) = vi
(xi(d⁎) + d⁎) is uniquely defined, for all i. Let x(d⁎) = y⁎.
5
By Lemma 1 and Lemma 1, the following corollary is immediate.
Corollary 1. (y⁎1 + d⁎, y⁎2 ,…, y⁎n ) as defined in Eqs. (12) and (13) forms
the unique equilibrium outcome of Γ N(1 + d⁎).
4. The limit result
We now increase the size of the problem by replicating a one-pie,
n-player problem k times. That is, in a k-replicated problem we allow
each replica of n players to bring a pie of size 1 to the pool of shareable
pies, and the resulting set of k · n players bargain over the resulting pie
of size k according to the procedure specified in the previous section.
Formally, let N = {1,…, n} be a set of original agents, and relabel
them by {11, 12, …, 1n}. Let the k times replicated – or k-replicated – set
of agents be {11,…,1n, 21,…,2n,…k1,…,kn}. That is, the k-replicated
problem contains k agents of type i a N, each with the preferences
of i. Attaching the player li the index h(li) = n · (l − 1) + i, we may
order players 11, …, kn according to their h-indices {h(11),…, h(kn)} =
{1,…n · k}. Using this indexation of the players, we specify a game
Γ{1,…,n ·k}(k), for any k = 1, 2, …. Then Propositions 1 and 2 are valid
for any k-replicated problem.4
By Proposition 1, the equilibrium of the k-replicated problem is
characterized by x(k)a Sk · n (k) and d(k) N 0 meeting Eqs. (3) and (4).
By symmetry, the following result is immediate:
By Eq. (10), d(kt)→ei(yi). By Eq. (14),
n
dðkt Þ
∑ x 1i ðkt Þ ¼ 1−
:
kt
i¼1
5
Given that d(kt)→ei(yi), we have ∑in= 1 x1i(k)→1.
Now we give a characterization of the unique convergence point of
x(k) on the Pareto frontier. More generally, the efficient n-vector y⁎
specifies how the gains of each generation are distributed among the
members of the generation when the economy grows large. This is our
main result.
Proposition 3. x1·(k) converges to y⁎ as specified in Eqs. (12) and (13)
when k tends to infinity.
Proof. Since, by Lemma 3, sequence {x1·(k)}∞
k = 1 is bounded, it suffices
to show that every convergent subsequence of it converges to y⁎. Let
subsequence {x1·(kt)}∞t = 1 converge to y. By Lemma 4, ∑in= 1 yi = 1. There is
a d N 0 such that yi = vi(yi + d) for all i = 1,…,n. By Lemma 1, y = y⁎ and
d = d⁎.
5
Thus, by Lemma 2, all sequences {xl·(k)}, for l = 0,1,…, converge to
y⁎ = (y⁎
1 ,…,y⁎)
n which is characterized by the data of the original n
players as in Eqs. (12) and (13).
Fig. 1 depicts how the limit outcome of a single replica is specified
in the n = 2 case. For any d N 0, identify the function v1(1 + d − x2) = x1 on
x2 a [0,1], and the function v2(1 + d − x1) = x2 on x1 a [0,1]. The unique
intersection (y1, y2) of the two functions satisfies
v1 ðð1 þ d−y2 Þ ¼ y1 ;
v2 ð1 þ d−y1 Þ ¼ y2 :
Then d⁎ chosen such that the intersection (y⁎
1 , y⁎
2 ) satisfies y⁎
1+
y⁎
2 = 1. Given such d⁎,
v1 1 þ dT −yT2 ¼ v1 yT1 þ dT ¼ yT1 ;
T
T
T
T
v2 1 þ d −y1 ¼ v2 y2 þ d ¼ yT2 :
Thus (y⁎
1 , y⁎
2 ) and d⁎ satisfy the conditions inn Eqs. (3) and (4) of a
two player game with X = 1 + d⁎.
Lemma 2. xli(k)= x(l + 1)i(k), for all i a N, for all l a {1,…,k}, for all k = 1, 2,….
Because of Lemma 2, it is sufficient to focus on x1·(k) =(x11(k),…,x1n(k)).
We may rewrite Eq. (4), for all k a {1, 2,…},
x 1i ðkÞ 0:
ð14Þ
dðkÞ ¼ k 1−∑ni¼1 Let {x(k)}∞k = 1 be a sequence of points meeting Eqs. (3) and (4) for
the respective k-replicated problems, for all k.
Lemma 3. Sequence {x1·(k)}∞
k = 1 is bounded.
∞
Proof. If {x1·(k)}∞
k = 1 is not bounded, there is a subsequence {x1·(kt)}t = 1 and
j such that x1j(kt)→∞. But given x1i ≥0 for all i, this would violate the
budget constraint (14).
5
By Lemma 3, {x1·(kt)}∞
k = 1 has a convergent subsequence.
4
Any indexation of the players would do.
Fig. 1.
252
K. Kultti, H. Vartiainen / Economics Letters 101 (2008) 249–252
5. Market for the first-proposer right
To conclude, we give a “Walrasian” interpretation to the characterized limit outcome y⁎. Being the first proposer in the bargaining
game is valuable. Consider a market where an arbitrator sells the right
to be the first proposer in a bargaining game to one of the n bargainers
and, once one player i, has paid price p for the right, adds p to the pool
of resources over which bargaining takes places. That is, given the
original size 1 of the pie, the player i becomes the first proposer in the
bargaining game Γ n(1 + p).
We study Walrasian markets for the first proposer right. We
concentrate on perfectly competitive equilibria. Inspired by Makowski
and Ostroy (2001), we see perfect competition as a situation where
each agent gets exactly what he brings to the economy.5 By Makowski
and Ostroy (2001), this holds if the agents are price takers and the
demand is perfectly elastic: given price p there are buyers that are
indifferent between buying or not buying. That is, in our set up the
price p is such that for at least two buyers the payoff from buying the
right with price p is equal to the payoff of letting the other buyer buy
the right at price p.
We claim that d⁎ is the unique such price and y⁎ is the resulting
allocation of the pie.
Proposition 4. e(y⁎) is the unique perfectly competitive Walrasian price
under which the players are indifferent, and y⁎ is the resulting allocation
of the pie, for d⁎ and y⁎ as specified in Lemma 1.
Proof. Let zj(1 + p) be what a non-proposer j gets in the game Γ n(1 + p).
Then the proposer i's share is 1 + p − ∑j ≠ i zj(1 + p). Since only one player,
say i again, eventually becomes the first proposer, there is a j ≠ i who is
indifferent between buying or not. Thus j's payoff from buying the
proposing right under p equals the buying cost p and the alternative
cost zi(1 + p), i.e.,
1 þ p− ∑ zk ð1 þ pÞ ¼ p þ zj ð1 þ pÞ:
k≠j
5
This is called full appropriation.
Then 1 = ∑in= 1 zi (1 + p). By Eqs. (3) and (4) (with X = 1 + p),
zi ð1 þ pÞ ¼ vi ðzi ð1 þ pÞ þ pÞ; for all i ¼ 1; :::; n:
5
By Lemma 1, zi(1 + p) = yi⁎ for all i, and p = d⁎.
By Proposition 3, the unique outcome y⁎ of the Walrasian market
for the first-proposing right can be thought as the expected outcome
of bargaining when the number of bargainers grows large and the
probability of a particular player having the right be the first proposer
becomes negligible. Having a large set of players is attractive since the
resulting bargaining outcome reflects strong average fairness: all but
one generation distribute their resources without a first mover
distortion. Thus the simple market game with small number of
players can be used to simulate the distortion-free many-player
bargaining outcome.
References
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